conference proceeding of iesa-2014.pdf

viii

International Conference on Industrial

Engineering Science and Applications-2014

Contents Paper ID Title of Papers Page No.

02 Progressive Failure Analysis of Laminated Composite Shells A Review 001-006

05 Application of Neural Network for Identification of Cracks on Cantilever Composite

Beam

007-011

06 Dynamic response of a simply supported beam with traversing mass 012-014

08 Mathematical modeling of steady state temperature distribution due to heat loss

from weld bead of a square butt joint

015-020

10 Computational Analysis of Shell Fluid of Shell and Tube Heat Exchanger Allowing

the Outcome of Baffles Disposition on Fluid Flow

021-025

11 Thermal Analysis of Porous Fin with Internal Heat Generation 026-031

13 Reliability Analysis of Two Lathe Machines Arranged in a Machining System 032-036

15 Genetic algorithm based performance analysis of 3-phase self-excited induction

generator

037-041

16 Comparative Study of Machining Processes by Process Capability Indices 042-045

17 Heat transfer analysis in porous fin of different profiles using Vibrational iteration

method

046-051

19 Patient Information implantation and reclamation from compressed ECG signal by

LSB watermarking technique

052-057

20 Vibration Characteristics of Rotating Simply Supported Shaft in Viscous Fluid 058-063

21 Diagnosis of Damage in Composite Beam Structures using Artificial Neural

Network with Experimental Validation

064-068

22 Finite Element Analysis of Hip Prosthesis for Identification of Maximum Stressed

Zone

069-072

23 Multi Channel Personal Area Network(MCPAN) Formation and Routing 073-079

24 Vibration analysis of a cracked Timoshenko beam 080-084

25 Silicon on Insulator based Directional, Cross Gap and Multimode Interference

Optical Coupler design

085-090

26 An Analysis of Short Term Hydrothermal Scheduling using Different Algorithms 091-096

28 Monitoring of the lung fluid movement and estimation of lung area using Electrical

Impedance Tomography: A Simulation Study

097-100

30 Multi-objective design of realistic load frequency control system using particle

swarm optimization

101-105

31 Hybridizing DE with PSO for Constrained Engineering Design Problems 106-111

33 Harmonic Distortion Optimization of Generalized A-Symmetrical Series/Parallel Multilevel Converter with Fewer Switches

112-117

34 Detailed study and proposed restoration of damaged structural bracket supports for

three tier insulated piping system by using anchoring methodology in filter house

structure of solvent dewaxing unit

118-122

39 Design and Development of a Heating- Cooling Belt using Thermoelectric

Refrigeration for Medical Purposes

123-127

42 Power line filter design considering losses and parasitic characteristics of passive

lumped components

128-133

43 CFD analysis of cambered airfoil for H-rotor VAWT 134-138

ix



45 Modeling of active transformation of microstructure of two-phase Ti alloys during

hot working

139-144

47 Value Based Planning of Renewable DGs in Distribution Network Incorporating

Variable Power Load Model and Load Growth

145-150

49 Phase Angle Measurement using PIC Microcontroller with Higher Accuracy 151-154

50 Automatic Electronic Water Level Management System using PIC Microcontroller 155-158

55 Effect of Temperature on Photovoltaic Cell performance 159-161

57 Optimum Process Scheduling Using Genetic Algorithm in an Existing Machine

Layout

162-166

59 A review parametric performance of solar still 167-172

61 Differential Difference Current Conveyor (DDCC) Based Current mode Integrator

and Differentiator

173-176

62 Performance analysis of 2 stroke gasoline engine by using compressed air 178-181

63 Design of Wide Band Digital Integrator and Differentiator 182-185

66 Tracking Mobile Targets Through Wireless HART 186-190

67 Hysteresis Compensation using Modified Internal Model Control for Precise Nano

Positioning

191-196

68 Reversible Data Hiding using Wavelet Transform and Compounding for DICOM

Image

197-201

69 Study on Photovoltaic System for Isolated and Non-Isolated Source Cascaded Two

Level Inverter (CTLI)

202-206

70 Mass Measuring System Using Delay-and-Add Direct Sequence (DADS) Spread

Spectrum Method

207-210

71 Fault analysis of wind generator connected power system using differential equation

technique

211-215

72 Biomedical application using zigbee 216-218

73 Resonant Frequency of 300-600-900 Right Angle Triangular Patch Antenna with

and without Suspended Substrate

219-222

74 Model Free Adaptive Control in Industrial Process: An Overview 223-226

76 Simulation of IGBT fed Mirror Inverter based H.F. Induction Cooker 227-230

82 A Hybrid Intelligent Algorithm Applied in Economic Emission Load Dispatch

Problems

231-235

86 Performance assessment of power system by incorporating Distributed generation

and Static VAR compensator

236-241

87 A study of Performance Analysis on Multi-bus Power Grid Network Modeling 242-246

89 Reliability Assessment of Energy Monitoring Service for a Futuristic Smart City 247-252

92 Performance Characteristics of 2x50kwp Roof top PV Power Plant System 253-257

93 Multiple Distributed Generator allocation by modified novel power loss sensitivity

for loss reduction

258-262

94 A Novel Constraint Increasing Approach for solving Sudoku puzzle 263-267

97 Kinematic synthesis of six bar gear mechanism 268-272

98 A Simulation Based Geometrical Analysis Of MEMS Capacitive Pressure Sensors

for High Absolute Pressure Measurement

273-277

100 RAHSIS: A Tool for Reliability Analysis of Hardware Software Integrated Systems 278-283

101 Steel Fiber Reinforced Concrete in the service of Civil Engineering 284-289

102 Optimal Control of singular systems via Haar function 290-293

x



103 Baseline Wander and Power Line Interference Removal in ECG Signal 294-297

105 Prediction of relative density of clean sand: A support vector machine approach 298-301

107 Strengthening of Structures using Glass Fibre Reinforced Plastic 302-306

109 Removal of sodium do-decylsulfate from waste water using adsoption on citrus

lemettioides

307-310

110 Design of Adiabatic Adder Structures for Low Power VLSI & DSP Applications 311-314

114 Variable Frequency Drives - a successful mode of speed control of AC motors 315-319

115 Using Appointment system to improve the loading process of trucks in a steel plant:

A simulation based case study

320-324

116 Productivity Improvement Through Line Balancing using Simulation 325-329

117 A Genetic Algorithm Trained Artificial Neural Network Based Selective Harmonic

Elimination Technique for Cascade Multilevel Inverters

330-335

118 Performance Analysis of Genetic Algorithm in Direction of Arrival of Wideband

Sources Over Wide SNR Range

336-339

119 Intelligent Co-ordinated Control for Boiler Turbine Unit 340-343

120 Vulnerability Assessment of Reinforced Concrete Buildings having Plan Irregularity

using Pushover Analysis

344-348

123 A comparative analysis of performance of three phase four wire DSTATCOM

topologies for power quality improvement

349-352

126 How Ergonomics play an important role in productivity improvement of an

organization

353-358

127 Modeling and Simulation of YNVD Transformer for Single Phase Electrified

Traction System

359-362

128 Transient Response and Load Sharing Improvement in Islanded Microgrids 363-367

129 Intelligent Hybrid Fuzzy PD Control for Trajactory Tracking of Robot Manipulator

and Comparative Analysis

368-372

130 Improvement in DC- link Voltage of Doubly Fed Induction Generator using SMES 373-377

131 Implementation of energy storage and FACT device with renewable power

generation system

378-381

132 Flexible pavement cost modeling for weak subgrade stabilized with fly ash and lime 382-386

133 Prediction of compression index of clay using artificial neural network 387-390

134 On the Directivity and Multiband Characteristics of Sierpinski Fractal on Bowtie 391-396

136 Power Quality Improvement Using DPFC Under Fault Conditions 397-401

137 Benchmarking and Analysis of the User-Perceived Performance of EPICS based

ICRH DAC

402-405

140 Power Flow Control in Smart Micro Grid using Fuzzy Controllers 406-409

144 Harmonics Mitigation with the help of Zsource Inverter based DVR 410-414

146 Assessment of Retailers quality in Dairy Supply Chain Using AHP Technique 415-419

149 Effect of Conflicting Vehicles on Service Delay Under Mixed Traffic Conditions 420-425

150 Defining Level of Service at Uncontrolled Median Openings through K-Medoid

Clustering

426-432

159 Optimum Design of FRP Rib Core Bridge Deck Panel using Gradient based

Optimization

433-439

160 Optimization design of FRP web core skew bridge deck system using Genetic

Algorithm

440-446

Progressive Failure Analysis of Laminated

Composite Shells A Review

Jayashree Sengupta

Post-Graduate Student: Department of Civil

Engineering

Jadavpur University

Kolkata, India

[email protected]

Dipankar Chakravorty

Professor: Department of Civil Engineering

Jadavpur University

Kolkata, India

[email protected]

[email protected]

AbstractComposite materials present striking potentials to

be tailored for advanced engineering applications. Thin walled

composite panels are one of the most utilized structural elements

in construction. The increasing use of the composites necessitates

for the precise and viable methods of analysis- the life prediction

being an important issue. The initiation and propagation of

failure until final fracture of the structure assesses the life of the

structure. Unnoticed internal failures may lead to fatal collapse,

thus making first ply and progressive failure of much concern to

researchers. This paper addresses the various literatures that

have been published so far associated with the progressive failure

of the composite laminated shells; also it reflects the five failure

theories working behind.

Keywordsprogressive failure; laminated composites; plates; shells; first-ply failure; literature review.

I. INTRODUCTION

Over the last few decades, composite laminates are

exceedingly used in various engineering sectors like

construction, mechanical, aerospace and marine; their

advantages being their high strength and stiffness to weight

ratio, extended fatigue life and various other superior material

properties. Unlike isotropic materials, the composites bear a

complex response to loadings which can be analysed now by

FEM.

The failure mechanisms are best understood at micro level.

However on a macroscopic level, the failure analysis is more

intricate. Upon the application of loads, the laminate

undergoes stresses closely related to the properties of the

constituent phases, i.e., matrix, fibre, and interface-interphase.

The first ply failure occurs when stresses in the weakest

lamina exceed the allowable strength of the same changing the

material properties. For a composite construction it is thus

crucial to locate this change. A composite material undergoes

transition in multiple phases. At first-ply failure, redistribution

of stresses occurs within the remaining laminae of the

laminate. This does not necessarily mean that the whole of

lamina has undergone failure; it only indicates the initiation.

The laminate will be termed as damaged with degraded

properties. The constitutive relations are changed followed by

reduction in stiffness. The stiffness of the failed lamina is not

taken into account and the rest are considered to remain

unaffected. The remaining laminae continue to take up load

till the ultimate strength is reached. A ply-by-ply progressive

analysis and the damage so done is analysed by the inclusion

of different failure criteria which allows for the identification

of the location of the failure.

II. LITERATURE REVIEW

Tsai and Wu [1] were the first to present that an

operationally simple strength criterion cannot possibly explain

the actual mechanisms of failures. Failures are but a multitude

of independent and interacting mechanisms. They made use of

strength tensors fulfilling the invariant requirements of

coordinate transformation; interaction terms were treated as

independent components and the difference in strengths owing

to positive and negative stresses were accounted for making it

way too improved than the Hill criterion wherein the

interactions were not independent of each other. Previously,

most initial failure analyses was concerned with the in-plane

loading cases, perhaps because of being more governing in

laminated structural elements. But Turvey [2] focused his

study where flexural load dominated limiting his research to

high modulus GFRP and CFRP and cross-ply symmetric

configurations. He considered Tsai-Hill failure criterion as it

is in good correlation for the GFRP laminates; expressed both

the deflection surface and lateral pressure deflection in Navier

double series form. Though his research was unrestricted on

grounds of loading, he however chose three cases viz.

uniformly distributed load, patch loading and hydrostatic

loading varying linearly.

Reddy and Pandey [3] studied the first ply failure for the

laminated composite plates, subjected to in-plane and/or

transverse loading, the first order shear deformation theory

and a tensor polynomial failure criterion (emphasis was laid

on maximum stress, maximum strain, Tsai-Hill, Tsai-Wu and

Hoffmans criteria) to predict failure at elemental Gauss points. They inferred that when the laminate was subjected to

in-plane loading all the failure criteria were capable of

predicting failure. However when the same was subjected to

transverse loading, maximum strain and Tsai-Hill happened to

have different results. The procedure that they depicted was an

iterative one; once the first-ply failure had occurred the load

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was reduced by 20% and the process went on till the

difference between any two successive failure loads was less

than 1%. They extended their research on post first ply failure

emphasizing on the progressive analysis [4]. Tolson and

Zabaras [5] studied progressive failure in laminated composite

plates implementing seven degrees of freedom (three

displacements, two rotations of normal about the plate mid-

plane, and two warps of the normal). The in-plane stresses

were calculated from the constitutive equations, but the

transverse stresses were assessed from the three-dimensional

equilibrium equations. The normal stress distributions were

calculated at each Gauss points. Thereafter the stresses were

employed in five failure criteria (maximum stress, Lee,

Hashin, Hoffman, and Tsai-Wu) and were checked against

whether failure had occurred or not. The Lee criterion gave

the best results.

Prior to this, Engblom and Ochoa [6] carried out PFA till

last ply failure; stiffness decrement and damage growth

followed standard laminate analysis. Analysis was carried out

on a plate subjected to uniaxial tension and four-point

bending. Chang and Chang [7] developed the progressive

damage analysis of notched laminated composites subjected to

tensile loading. The progressive failure method used a

nonlinear FEA using the modified Newton-Raphson iteration

scheme to work out the state of stress in a composite plate.

Chang and Lessard [8] studied the damage in laminated

composites containing an open hole subjected to compressive

loading, wherein the in-plane response of the laminates from

initial loading to final collapse was studied. A geometrically

non-linear formulation based on finite deformation theory was

used. Reddy and Reddy [9] computed linear and non-linear

first-ply failure loads of composite plates for different load

cases and edge conditions. The linear loads results varied by a

maximum of 35% and for non-linear loads it was 50%.

Besides, this difference was much large for thin laminates

subjected to transverse loading and quite small for thick

laminates subjected to in-plane loadings. They extended their

work [10] to nonlinear progressive analysis using the

Generalized Laminate Plate Theory (GLPT) of Reddy

applying a new stiffness reduction format.

Kim and Hong [11] studied macroscopic failure models

evaluating the stiffness changes employing shear lag factor

and fiber bundle failure. Tan and Perez [12] studied

progressive failure of laminated composites with holes,

subjected to compressive loading predicting the extent of

damage at any level of loading. Results showed good

assessment with the experimental results. Tan [13] had

previously developed progressive failure model of laminated

composites subjected to in-plane loading considering the

environmental impacts (thermal and hygroscopic stresses).

Kam and Sher [14] studied progressive failure of centrally

loaded laminated composite plates. The Ritz method, with

geometric non-linearity, in the Von Karman sense, was used

to construct the load displacement behaviour. Cheung et al.

[15] presented a PFA of composite plates by the finite strip

method based on higher order shear deformation theory and

Lees failure criterion. Sahid and Chang [16] developed a model for predicting the effects of matrix crack induced

accumulated damage on the in-plane response of laminated

composites under tensile and shear loads. Echaabi et al. [17]

presented a theoretical and experimental study of damage

progression and failure modes of composite laminates under

three point bending.

Kim et al. [18] carried out PFA to predict the failure

strengths and failure modes (tension, shear-out and bearing)

which were judged against experimental data, of pin-loaded

laminated composite plates using the penalty finite element

method. Hashins failure criteria was performed for damage evaluation in the laminates. Gummadi and Palazotto [19]

performed PFA of composite cylindrical shells with large

rotations based on Langrangian approach and the due changes

in the constitutive relations were discussed; considered failure

modes of matrix cracking, fiber breakage and delamination;

damage progression followed maximum stress criterion. The

stiffness matrix was determined based on Greens strain and 2nd Piola Kirchoff stresses. Greens strain was transformed to Almansi strain which was further transformed into material

axis system and was used to determine the Euler stresses.

These Euler stresses were the key to failure determination.

Padhi et al. [20] presented their detailed study on progressive

failure and ultimate collapse of laminated composites using

Hashin and Tsai-Wus failure criteria. They made use of Newton-Raphson Method with a force and moment residual

convergence of 0.5% and displacement correction

convergence of 1%. The model was capable of assessing type

and extent of damage all throughout.

Spottswood and Palazotto [21] determined the response

together with material failure of a thin curved composite shell

resisting transverse loading, incorporated simplified large

displacement/rotation (SLR) theory and compared the results

with previously published available data; failure criteria being

Hashins. Pal and Ray [22] carried out PFA under transverse static loading in linear elastic range distinctively for both

antisymmetric and symmetric angle ply laminates. Knight Jr.

et al. [23] reviewed the overall computational issues and

requirements for performing PFAs using STAGS (Structural

Analysis of General Shells) for solving non-linear quasi-static

structural response problems including special details. Prusty

[24] studied unstiffened and stiffened composite panels under

transverse static loadings in the linear elastic range. Eight-

noded isoparametric quadratic elements with three-noded

curved beam elements were modeled and checked against

various failure theories. Ultimate ply failure loads for the

stiffened panels with cross-ply and angle-ply laminations in

the shell was analyzed.

Akhras and Li [25] proposed a spline finite strip method

PFA of thick composite plates based on Chos higher order zigzag laminate theory and Lees failure criterion. The transverse shear stresses were obtained directly from the

constitutive equations; the shear correction factor was not

required as for the first-order shear theory. The procedure

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involved an incremental load analysis through modified

Newton-Raphson method and standard PFA method showing

good agreement with the 3D finite element solutions. Las and

Zemck [26] proposed a PFA model of unidirectional

composite panels using Pucks fiber and inter-fiber criteria demonstrating the results with examples of tensile tests of

single-ply panels. Tapered laminated plates under the action of

uniaxial compression were investigated by Ganesan and Liu

[27] predicting different failure loads and the associated

displacements, locations and modes. The influences of the

tapered configuration, lay-up configuration, and fiber

orientation were also the concern of the study. Earlier,

Ganesan and Zhang [28] had conducted a detailed

investigation of the progressive failure of uniform thickness

laminates subjected to uniaxial compression. Singh et al. [29,

30, 31] also studied the progressive failure of uniform

thickness plates subjected to different loads. Progressive

damage analysis methodology for stress analysis of composite

laminated shells using finite strip methods based on Mindlins plate-bending theory were addressed by Zahari and El-Zafrany

[32], where the non-linear equations were derived using the

tangential stiffness matrix approach; validation was done by

comparing the results with analysis in ABAQUS.

Sandwich composite panels under quasi-static impact were

investigated by Fan et al. [33]. Hashins and Besants criteria were checked for different failure mechanisms. Ply

discounting method was employed as the strategy for material

degradation. Ahmed and Sluys [34] designed a computational

model presenting a mesoscopic failure model studying matrix

cracking, delamination and the combined effect. A mesh

independent matrix cracking was modeled with discontinuous

solid-like shell element (DSLS); delamination was presented

by a shell interface model. Besides material nonlinearities, the

numerical model simulated geometrical nonlinearities.

Anyfantis and Tsouvalis [35] studied post-buckling

progressive and final failure response of stiffened composite

panels using ANSYS. The intralaminar, fiber and matrix

failure modes in compression and tension were addressed

using a combined framework of Hashins and Tai-Wu failure. The PFA method included intralaminar failures that

stimulated material degradation of the failed layers.

Pietropaoli [36] too worked on progressive failure of

composite structures using a constitutive model implemented

in ANSYS. Standard ply discount technique was used and the

onset and progression of damage was observed and the results

were validated against experimental results. Bogetti et al. [37]

studied the nonlinear response and progressive failure of

composite laminates under tri-axial loading. A program was

build and executed in MATLAB; this analyzed and displayed

the failure envelope. Philippidis and Antoniou [38] computed

a PFA model for glass/epoxy composite giving an extensive

comparison between numerically calculated stressstrain response up to failure and experimental data. Crdenas et al.

[39] presented a reduced-order FE model suitable for PFA of

composite structures under dynamic aeroelastic conditions

based on a Thin-Walled Beam theory predicting both onset

and propagation of damage.

III. FAILURE CRITERIA

A. Notations

FX Overall Longitudinal Strength

FY Overall Transverse Strength

FXT , FYT Tensile strength in X and Y direction

respectively

FXC , FYC Compressive strength in X and Y direction

respectively

FS In-Plane shear strength

, Normal stresses in X and Y direction

respectively

Shear stress in X-Y plane

, Strain along X and Y direction respectively

Shear strain

,

Ultimate Strain along X and Y direction

respectively

Ultimate Shear strain

Generally failure criteria can be either non-interactive

(independent) or interactive (polynomial). An independent

criterion gives the mode of failure, be it longitudinal or

transverse, tensile or compressive or shear mode, and is

simple to apply. However, the effect of stress interactions is

ignored. The stress interactions are addressed by the

polynomial failure criteria; but in this case, the failure mode is

disregarded. The laminate may indicate failure using a non-

interactive theory. If not so, the lamina should be checked

using the interactive failure. It may so happen that the

independent stresses do not initiate failure but their

interactions may. Hence it is best to check for failure through

both independent and non-interactive criteria.

B. Independent Failure Criteria

1) Maximum Stress Criteria: According to maximum stress theory, the failure initiates if at least one of the criteria

is satisfied,

1,

1,

1 (1)

2) Maximum Strain Criteria: According to maximum strain criteria failure occurs if any of the following is reached,

1,

1,

1 (2)

C. Interactive Failure Criteria

Failure can be predicted by the following tensorial form,

1 (3)

which when expanded in two dimensional form gives,

1

(4)

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The failure indices for different theories are as follows.

TABLE I. FAILURE INDICES

Failure

Indices

Tsai

Hill(a) Tsai Wu Hoffmann

(a). If > 0, = FXT else = FXC . Similarly if > 0, = FYT else = FYC

IV. PROGRESSIVE FAILURE ANALYSIS

To carry out PFA, it is necessary to determine the first ply

failure of the laminate. The material properties of the lamina,

mechanical loading i.e. forces and moments, and layer

orientation are read. Based on the inputs, the stresses at each

Gauss point within individual lamina were evaluated and are

verified with the failure criteria to check any possible failure.

If any failure had occurred, the material properties at that

point were modified with accordance to the observed failure

mode, and the stresses were recalculated at FPF load, using a

property degradation technique- (A) Total-Ply Failure Method

and (B) Partial-Ply Failure Method. Thereafter, a check is

performed to see whether the second ply fails at FPF load, if

not, a load increment is performed and with the reduced

stiffness the process continues till the ultimate failure load

occurs and at that stage convergence of stresses cannot be

achieved numerically . A typical PFA is demonstrated in Fig.

1.

A. Total Ply Failure Method

On reaching the failure, the strength and stiffness of the

failed ply is totally reduced to zero. This implies that if the ply

undergoes matrix failure, it is no longer able to carry load in

fibre direction, which, may not be the case. Thus this method

somehow underestimates the laminate strength.

B. Partial Ply Failure Method

In this approach, the failure mode is taken into account. If

the ply fails due to fibre failure, the stiffness of the failed ply

is reduced to zero. However, if it is a matrix controlled failure

or shear failure, the longitudinal modulus retains its value but

the transverse and shear modulus are set to zero.

Fig. 1. Flowchart of Progressive Failure Analysis Methodology

V. CONCLUSION

Various literatures on laminated composite structures and are

studied in this paper. It is visibly evident that prediction of the

failure process, the initiation and growth of the damages, and

the maximum loads that the structures can withstand before

failure occurs is essential for assessing the performance of

composite laminated plates and for developing reliable and

safe design. It is found that such studies on beams and plates

have appeared in quite a number of places although similar

studies on composite shells are really scarce. Hence failure

analyses of composite shells need careful attention.

Cylindrical and spherical shells enjoyed quite an importance;

however their evaluation on progressive failure is still an area

of interest. Industrial shell forms like conoids and skewed

hypar need attention as well. These shells are ruled, doubly

curved, aesthetically appealing and easy to cast and fabricate. Hypar shell has wide applications in engineering and was studied

by a number of researchers like Kielb et al. [40], Seshu and

Ramamurti [41] and Qatu and Leissa [42]. Researchers like Das

and Chakravorty [43-46, 48] and Kumari and Chakravorty [48,

49] also studied behavioral characteristics of laminated conoidal

shells. These shell forms are to be studied for failure also.

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[23] Norman F. Knight Jr., Charles C. Rankin, Frank A. Brogan, STAGS computational procedure for progressive failure analysis of laminated

composite structures, International Journal of Non-Linear Mechanics Vol.37,

2002, pp. 833849

[24] B. Gangadhara Prusty, Progressive Failure Analysis of Laminated Unstiffened and Stiffened Composite Panels, Journal Of Reinforced Plastics And Composites, Vol. 24, No. 6, 2005 pp.633-642.

[25] G. Akhras and W.C. Li Progressive failure analysis of thick composite plates using the spline finite strip method, Composite Structures Vol.79, 2007, pp.34-43

[26] Vladislav Las and Robert Zemck , Progressive Damage of Unidirectional Composite Panels, Journal of Composite Materials, Vol. 42, No. 1, 2008 pp. 25-44

[27] Rajamohan Ganesan and Dai Ying Liu, Progressive failure and post-buckling response of taperedcomposite plates under uni-axial compression, Composite Structures, Vol.82, 2008, pp. 159176

[28] R. Ganesan and D.Zhang, Progressive failure analysis of composite laminates subjected to in-plane compressive and shear loadings, Science and Engineering of Composite Materials, 2004, Vol.11 (23), pp. 79-102.

[29] SB Singh, A Kumar and NGR Iyengar, Progressive failure of symmetrically laminated plates under uni-axial compression. Structural Engineering and Mechanics 1997, Vol.5, pp. 433-50.

[30] SB Singh, A Kumar and NGR Iyengar, Progressive failure of symmetric laminates under in-plane shear: I-Positive shear., Structural Engineering and Mechanics, 1998, Vol.6(2),pp. 143-59.

[31] SB Singh, A Kumar and NGR Iyengar, Progressive failure of symmetric laminates under in-plane shear: II-Negative shear. Structural Engineering and Mechanics, 1998, Vol.6(7), pp. 75772.

[32] R. Zahari and A. El-Zafrany, Progressive failure analysis of composite laminated stiffened plates using the finite strip method, Composite Structures Vol.87, 2009, pp. 6370.

[33] X.L. Fan, T.J. Wang and Q. Sun Damage evolution of sandwich composite structure using a progressive failure analysis methodology, Procedia Engineering, Vol.10, 2011, pp. 530535.

[34] A. Ahmed and L. J. Sluys, A Computational Model For Prediction Of Progressive Damage In Laminated Composites, ECCM15 - 15th European Conference on Composite Materials, Venice, Italy, 24-28 June 2012.

[35] Konstantinos N. Anyfantis and Nicholas G. Tsouvalis, Post Buckling Progressive Failure Analysis of Composite Laminated Stiffened Panels, Applied Composite Materials, June 2012, Vol. 19, Issue 3-4, pp. 219-236

[36] Elisa Pietropaoli, Progressive Failure Analysis of Composite Structures Using a Constitutive Material Model (USERMAT) Developed and

Implemented in ANSYS, Applied Composite Materials, June 2012, Vol. 19, Issue 3-4, pp. 657-668.

[37] Travis A. Bogetti, Jeffrey Staniszewski, Bruce P Burns, Christopher PR Hoppel Christopher PR, John W Gillespie Jr. and John Tierney, Predicting

the nonlinear response and progressive failure of composite laminatesunder

tri-axial loading., Journal of Composite Materials, Vol.46, Issue-19-20, 2012, pp. 24432459.

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[41] P. Seshu and V. Ramamurti, Vibration of twisted composite plates, Journal of Aeronautical Society of India, Vol. 41, 1989, pp. 65-70.

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[43] H. S. Das and D. Chakravorty, Design aids and selection guidelines for composite conoidal shell roofs A finite element application, Journal of Reinforced Plastics and Composites, Vol. 26, No. 17, 2007 pp. 1793-1819.

[44] H. S. Das and D. Chakravorty, Natural frequencies and mode shapes of composite conoids with complicated boundary conditions, Journal of Reinforced Plastics and Composites, Vol. 27, No. 13, 2008 pp. 1397-1415.

[45] H. S. Das and D. Chakravorty, Composite full conoidal shell roofs under free vibration, Advances in Vibration Engineering, Vol. 8, No. 4, 2009, pp. 303 310.

[46] H. S. Das and D. Chakravorty, Finite element application in analysis and design of pointsupported composite conoidal shell roofs suggesting selection

guidelines, The Journal of Strain Analysis for Engineering Design, Vol. 45, No. 3, 2010, pp. 165 177.

[47] S. Kumari and D.Chakravorty, On the bending characteristics of damaged composite conoidal shells a finite element approach, Journal of Reinforced Plastics and Composites, Vol. 29, No. 21, 2010, pp. 3287-3296.

[48] H. S. Das and D. Chakravorty, Bending analysis of stiffened composite conoidal shell roofs through finite element application, Journal of Composite Materials, Vol. 45, No. 5, 2011, pp. 525- 542.

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[51] Engineering Mechanics of Composite Materials by Isaac M. Daniel and Ori Ishai, Oxford University Press.

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Application of Neural Network for Identification of Cracks on Cantilever

Composite Beam

Irshad A Khan

Mechanical Engg. Department

National Institute of Technology

Rourkela, India

[email protected]

Adik Yadao



Rourkela, India

[email protected]

Dayal R Parhi



Rourkela, India

[email protected]

Abstract In the current analysis numerical and neural network methods are adopted for detection of crack in a

cantilever composite beam structure containing non propagating

transverse cracks. The presence of cracks a severe threat to the

performance of structures and it affects the vibration signatures

(Natural frequencies and mode shapes). The material used in this

analysis is glass-epoxy laminated composite. The numerical

analysis is performed by using commercially available software

package ANSYS to catch the relation between the change in

natural frequencies and mode shapes for the cracked and un-

cracked cantilever composite beam. Which subsequently used to

the design of smart system based on neural network for

prediction of crack depths and locations following inverse

techniques. The neural controller is developed with relative

natural frequencies and relative mode shapes difference as input

parameters to calculate the deviation in the vibration parameters

for the cracked dynamic structure. The output from the neural

controller is relative crack depth and relative crack location.

Results from numerical analysis are compared with experimental

results having good agreement to the results predicted by the

neural controller.

Keywords Crack; neural network; Natural frequencies; Mode shapes; Ansys.

I. INTRODUCTION

Health monitoring and the analysis of damage in the form of

crack in Beam like dynamic structures are important not only

for leading safe operation but also retraining system

performance. Since long efforts are on their way to find a

realistic solution for crack detection in beam structures in this

regard many approaches have so far being taken place. When

a structure suffers from damages, its dynamic properties can

change. Crack damage leads to reduction in stiffness also with

an inherent reduction in natural frequency and increase in

modal damping.

Discrete Wavelet Transform based method is presented for the

identification of multiple cracks in polymeric laminate

composite beam by Andrzej K., [1]. The valuation of the crack

locations is based on the estimation of natural mode shapes of

crack and uncrack beams. The mode shapes were estimated

experimentally using laser Doppler vibrometry. Krawczuk M,

et al. [2] proposed two models witch gives valuable

information about the location and size of defects in the

beams. This method makes it possible to construct beam finite

elements with various types of cracks (double edge, internal,

etc.) If the stress intensity factors for a given type of crack are

known. Damage identification on a composite cantilever beam

through vibration analysis using finite element analysis

software package ANSYS is established by Ramanamurthy et

al. [3]. Damage Algorithm and Damage index method used to

identify and locate the damage in the composite beam. A

composite matrix cracking model is implemented in a thin-

walled hollow circular cantilever beam using an effective

stiffness approach by Pawar et al. [4]. The composite beam

model is used to develop a genetic fuzzy system to detect and

locate the presence of matrix cracks in the structure.

Continuous wavelet transform is used to identification of

crack in beam like structures by analysing the natural

frequency and mode shape of cracked cantilever beam by Rao

et al. [5]. The effects of cracks on the dynamic characteristics

of a cantilever composite beam are studied by Kisa M. [6].

The material of composite beam is graphite fibre-reinforced

polyamide containing multiple transverse cracks. The effects

of the crack location and depth and the fibre volume fraction

and orientation of the fibre on the natural frequencies and

mode shapes of the beam are explored. Two Damage

identification algorithms are established for assessment of

damage using modal test data which are similar in concept to

the subspace rotation algorithm or best feasible modal analysis

method by Hu et al. [7]. Moreover, a quadratic programming

model is set up the two methodologies to damage assessments.

II. NUMERICAL ANALYSIS

The numerical analysis is brought out for the cracked

cantilever composite beam shown in fig1, to locate the mode

shape of transverse vibration at different crack depth and crack

location. The cracked beams of the current research have the

following dimensions.

Length of the Beam (L) = 800mm, Width of the beam (W) =

50mm, Thickness of the Beam (H) = 6mm

Relative crack depth (1=a1/H) varies from 0.0833 to 0.5;

Relative crack depth (2=a2/H) varies from 0.0.833 to 0.5;

Relative crack location (1=L1/L) varies from 0.625 to 0.875;

Relative crack location (2=L2/L) varies from 0.125 to 0.9375;

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Properties of Glass-Epoxy composite material in analysis:

Youngs modulus of fiber = Ef = 72.4 GPa; Youngs modulus

of matrix = Em = 3.45 GPa;

Modulus of rigidity of fiber = Gf = 29.67 GPa; Modulus of

rigidity of matrix = Gm = 1.277 GPa;

Poissons ratio = f = 0.22; Poissons ratio = m = 0.35;

Density of fiber = f = 2.6 gr/cm3; Density of matrix = m =

0.33 gr/cm3;

Numerical modal analysis based on the finite element

modeling is performed for studying the dynamic response of a

dynamic structure. The natural frequencies and mode shapes

are significant parameters in designing a structure under

dynamic loading conditions. The numerical analysis is

accepted by using the finite element software ANSYS in the

frequency domain and obtain natural frequencies, and mode

shapes.

A higher order 3-D, 8 node element having three degrees of

freedom at each node: translations in the nodal x, y, and z

directions (Specified as SOLSH190 in ANSYS) was selected

and used throughout the analysis. Each node has three degrees

of freedom, making a total twenty four degrees of freedom per

element. The layers stacking in ANSYS shown in fig2. The

results of the numerical analysis for the first three mode

shapes for un-cracked and cracked beam, having cracks

orientation 1=0.166, 2=0.333 and 1=0.25, 2=0.5 are shown

in the fig3.

Fig.2 Layers Stacking in ANSYS

Fig. 1 Geometry Cantilever beam with multiple cracks

Fig.3a. Relative Amplitude vs. Relative crack location from

fixed end (1st mode of vibration)

Fig.3c. Relative Amplitude vs. Relative crack location from


Fig.3b. Relative Amplitude vs. Relative crack location from


W

H

a2 L1

L2

L

a1

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III. NEURAL NETWORK ANALYSIS

A back-propagation neural controller has been developed for

detection of the relative crack locations and relative crack

depth having six input parameters and two output parameters

as shown in fig. 4. The term used for the inputs are as follows;

Relative first natural frequency = rfnf; Relative second

natural frequency = rsnf; Relative third natural frequency =

rtnf; Relative first mode shape difference = rfmd; Relative

second mode shape difference = rsmd; Relative third mode

shape difference = rtmd. The term used for the outputs are as

follows; Relative first crack location = rcl1 Relative second

crack location = rcl2 Relative first crack depth = rcd1

Relative second crack depth = rcd2

A. Neural controller mechanism for crack detection

The neural network used is a ten-layered perceptron. The

chosen numbers of layers are found empirically to facilitate

training. The input layer has six neurons, three for first three

relative natural frequencies and other three for first three

average relative mode shape difference. The output layer has

four neurons, which represents relative crack locations and

relative crack depths. The first hidden layer has 12 neurons,

the second hidden layer has 36 neurons, the third hidden layer

has 50 neurons, the fourth hidden layer has 150 neurons, the

fifth hidden layer has 300 neurons, the sixth hidden layer has

150 neurons, the seventh hidden layer has 50 neurons, and the

eighth hidden layer has 8 neurons. These numbers of hidden

neurons are also found empirically. Fig 5 depicts the neural

network with its input and output signals.

IV. EXPERIMENTAL INVESTIGATION

To validate the numerical analysis result, an experiment on composite beam has been performed shown in fig 6. A composite beam was clamped at a vibrating table. During the experiment the cracked and un-cracked beams have been vibrated at their 1

st, 2

nd and 3

rd mode of vibration by using an

exciter and a function generator. The vibrations characteristics such as natural frequencies and mode shape of the beams correspond to 1

st, 2

nd and 3

rd mode of vibration have been

recorded by placing the accelerometer along the length of the beams and displayed on the vibration indicator. The experimental results are in close justification with neural analysis results. These results for first three modes are plotted in fig7. Corresponding numerical results for the cracked and un-cracked beam are also presented in the same graph for comparison. The comparison of results between neural controller, numerical, experimental analysis shown in table1.

Fig.4 Neural model

Fig. 5 Multilayers feed forward back propagation neural model for

damage detection

1. Data acquisition (Accelerometer); 2. Vibration analyser;

3. Vibration indicator embedded with software (Pulse Labshop);

4. Power Distribution; 5. Function generator; 6. Power amplifier;

7. Vibration exciter; 8. Cracked Cantilever Composite beam

Fig.6 Schematic block diagram of experimental set-up

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V. CONCLUSION

The conclusions derived from the various analyses as

mentioned above are depicted below.

1. The Numerical analysis results are well agreed with

neural analysis results.

2. The investigation of vibration signatures of the

cracked and un-cracked composite beam shows a

variation of mode shapes and natural frequencies.

3. The numerical analysis and neural analysis results are

compared with the experimental results. They have

good judgment.

4. The present method can be engaged as a health

diagnostic tool for vibrating faulty structures.

5. Proposed health monitoring technique can be used for

composite as well as isotropic material.

VI. REFERENCES

[1] A. Katunin, Identification of multiple cracks in composite beams

using discrete wavelet transforms, Scientific Problems of

Machines Operation and Maintenance, vol. 2(162), pp. 41-52,

2010.

[2] M. Krawczuk, and W. M Ostachowicz, Modelling and vibration

analysis of a cantilever composite beam with a transverse open

crack, Journal of Sound and Vibration, vol. 183(1), pp. 69-

89,2005.

[3] E.V.V. Ramanamurthy, and K. Chandrasekaran, Vibration

analysis on a composite beam to identify damage and damage

Severity using finite element method, International Journal of

Engineering Scie and Techno, vol. 3, pp. 5865-5888, 2012.

[4] P. M. Pawar and R. Ganguli, Matrix Crack Detection in Thin-

walled Composite Beam using Genetic Fuzzy System, Journal of

Intelli Material Sys and Struct, vol. 16(5), pp. 395 -409, 2005.

[5] Rao, D.Srinivasa. Rao, K. Mallikarjuna, G.V. Raju, Crack

Identification on a beam by Vibration Measurement and Wavelet

Analysis, International Journal of Engineering Science and

Technology vol. 2(5), pp.907-912, 2010.

[6] M. Kisa, Free vibration analysis of a cantilever composite beam

with multiple cracks, Composites Science and Technology, vol.

64(9), pp. 1391-1402, 2004.

[7] N. Hu, X. Wang, H. Fukunaga, Z.H Yao, H.X. Zhang, Z.S.Wu,

Damage Assessment of structures using modal test data,

International Journal of solids and structures, vol. 38(18), pp.

3111-3126, 2001.

Fig.7a. Relative Amplitude vs. Relative crack location from


Fig.7b. Relative Amplitude vs. Relative crack location from

fixed end (2nd mode of vibration)

Fig.7c. Relative Amplitude vs. Relative crack location from

fixed end (3rd mode of vibration)

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Exp

erim

enta

l re

lati

ve

1st c

rack

dep

th

rcd

1

1st c

rack

loca

tio

n

rcl1

2n

d c

rack

dep

th

rcd

2,

2n

d c

rack

loca

tion

rcl

2 rc

l2

0.7

2

0.6

21

0.6

20

0.4

6

0.4

8

0.6

9

0.6

20

0.6

23

rcd2

0.4

10

0.2

2

0.2

1

0.4

12

0.2

2

0.1

8

0.4

5

0.2

3

rcl1

0.2

1

0.3

70

0.3

71

0.1

21

0.2

3

0.1

9

0.3

70

0.3

70

rcd1

0.2

1

0.1

63

0.1

62

0.3

29

0.1

63

0.4

4

0.3

27

0.4

12

Nu

mer

ical

rel

ativ

e

1st c

rack

dep

th

rcd

1

1st c

rack

loca

tio

n

rcl1

2nd c

rack

dep

th

rcd

2,

2nd c

rack

loca

tion

rcl

2

rcl2

rcl2

0.7

3

0.6

23

0.6

21

0.4

7

0.4

7

0.7

1

0.6

22

0.6

24

rcd2

0.4

12

0.2

4

0.2

2

0.4

13

0.2

1

0.1

9

0.4

6

0.2

4

rcl1

0.2

3

0.3

72

0.3

73

0.1

23

0.2

2

0.2

0

0.3

71

0.3

72

rcd1

0.2

2

0.1

64

0.1

63

0.3

31

0.1

62

0.4

6

0.3

29

0.4

13

Neu

ral

Con

tro

ller

rela

tiv

e1st c

rack

dep

th

rcd

1

1st c

rack

loca

tio

n

rcl1

2nd c

rack

dep

th

rcd

2,

2nd c

rack

loca

tion

rcl

2

rcl2

0.7

4

0.6

25

0.6

23

0.4

9

0.5

11

0.7

31

0.6

24

0.6

26

rcd2

0.4

15

0.2

42

0.2

42

0.4

14

0.2

33

0.2

1

0.4

8

0.2

6

rcl1

0.2

6

0.3

53

0.3

73

0.1

24

0.2

43

0.2

2

0.3

73

0.3

74

rcd1

0.2

6

0.1

76

0.1

63

0.3

34

0.1

64

0.4

8

0.3

21

0.4

26

Av

erag

e

Rel

ativ

e

thir

d

mo

de

shap

e

dif

fere

nce

tm

d

0.0

132

0.0

082

0.0

732

0.0

752

0.0

152

0.0

079

0.0

247

0.0

162

Av

erag

e

Rel

ativ

e

seco

nd

mo

de

shap

e

dif

fere

nce

sm

d

0.0

346

0.0

021

0.0

09

0.0

026

0.0

267

0.0

025

0.0

069

0.0

019

Av

erag

e

Rel

ativ

e

firs

t

mo

de

shap

e

dif

fere

nce

fm

d

0.0

036

0.0

017

0.0

126

0.0

012

0.0

048

0.0

017

0.0

065

0.0

046

Rel

ativ

e

thir

d

nat

ura

l

freq

uen

cy

tn

f

0.9

889

0.9

978

0.9

937

0.9

975

0.9

881

0.9

974

0.9

871

0.9

988

Rel

ativ

e

seco

nd

nat

ura

l

freq

uen

cy

sn

f

0.9

979

0.9

989

0.9

944

0.9

989

0.9

986

0.9

973

0.9

857

0.9

985

Rel

ativ

e

firs

t

nat

ura

l

freq

uen

cy

fn

f

0.9

987

0.9

997

0.9

958

0.9

981

0.9

981

0.9

989

0.9

980

0.9

993

Tab

le1

. C

om

par

ison

of

resu

lts

bet

wee

n N

eura

l co

ntr

oll

er, nu

mer

ical

and

exp

erim

enta

l an

aly

sis

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Dynamic response of a simply supported beam with

traversing mass Shakti.P.Jena

Mechanical Engineering Department


Rourkela, Odisha, India

[email protected]

Dayal R.Parhi Mechanical Engineering Department


Rourkela, Odisha, India

[email protected]

Abstract- In the present research, A theoretical- computational

technique has been developed to evaluate the dynamic response of a

simply supported beam subjected to a traversing mass with different

boundary conditions. The effects of speed of the moving mass on the

dynamic response of the beam have been investigated and the critical

influential speed (CIS) of the beam also found out. The governing

equation of motion of the structures has been converted into a series

of coupled ordinary differential equation and solved by fourth order

Runge-Kutta technique.

Keywords- Traversing mass, dynamic response, CIS, Runge-

Kutta Technique

Introduction : A traversing mass produces larger beam

deflections and stresses than that of the same mass applied

statically. Hence the analysis of structures subjected to

traversing mass is of important problem. These deflections and

stresses are the function of both velocity and time of the

moving mass.The dynamic analysis of structures under

moving loads has been an interesting subject of various

researches over the last decades.Its mainly due to

improvement of new high-speed vehicles and general drive to

improve the active communications.

Jeffcott [1] has first developed a technique to get the

inexact solution for the problem of vibration with the action of

moving and variable loads. Stanisic et al. [2] developed a

theory to calculate the response of a plate to a multi-masses

moving structure by Fourier transformation technique. The

analysis shows that for the same natural frequency of the plate,

the resonance is reached prior by considering the moving

multi-masses system than by moving multi-forces system.

Stanisic and Hardin [3] have examined to find out the response

of a simple supported beam subjected to random no of moving

masses by Fourier technique in the existence of external load.

Stansic and Euler [4] have investigated an article to describe

the dynamic response of the structures with moving masses

using operational calculus method. Akin and Mofid [5] have

developed an analytical-numerical method to compute the

dynamic response of beams subjected to moving mass with

arbitrary boundary states by variable separable technique and

compared the results to finite element analysis (FEA). Olsson

[6] has investigated the dynamic response of a simply

supported beam with a continuous traversing load at constant

velocity. Parhi and Behera [7] have investigated the dynamic

deflection of a cracked beam with moving mass by Runge-

Kutta technique and the energy method.

Ichikawa et al. [8] have investigated the response of a multi

span Euler-Bernoulli beam carrying moving mass by using

Eigen function technique and direct integration method

combinedlly. Siddiqui et al. [9] have analyzed the dynamic

characteristics of a flexible cantilever beam with moving mass-

sprung system by using perturbation, time-frequency and

numerical methods. Azam et al. [10] have investigated the

response of a Timoshenko beam under moving mass and

moving sprung structures. Bilello et al. [11] have developed a

scale-model experiment to analyze the response of a mass

traversing on a Euler-Bernoulli beam by the application of

theory of structural models. Lee and Renshaw [12] have

investigated a new solution method to solve the moving mass

structures problem for non conservative, linear, distributed

parameter systems using complex Eigen function expansions.

Sniady [13] has studied the dynamic response of a simple

supported Timoshenko beam acted upon by a force traversing

with constant speed. Esmailzadeh and Ghorashi [14] have

studied the reaction of a Timoshenko beam subjected to a

constant partly disseminated travelling mass. Dehestani et al.

[15] have analyzed the necessity of Coriolis acceleration

associated with the moving mass during the motion along the

vibrating beam and the response of Timoshenko beam along

with moving mass by introducing the coefficient of influential

speed.

In the present paper, the dynamic deflection of a simple

supported beam with moving mass has been determined and

the consequences of speed on the beams response also found out. The analysis has been carried out with constant mass

magnitude at different velocities.

Problem Formulation:

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Timet in Sec (Hinged-Hinged Beam)

Dif

lecti

on

at

the m

idsp

an

(mm

)

Considering a simply supported beam subjected to a moving

mass M with velocity V. It is assumed that during the

course of motion there is no separation between the moving

mass and the beam .So the transverse of the moving mass with

respect to the beam is ignored here.

Nomenclature:

L=Beam Length, m=Beam mass per unit length, M=Mass of

the moving object, V (t) =Velocity of the moving mass with

respect to time, I=Constant moment of inertia, =Distance travelled by the traversing mass from the fixed end.

y(x,t)=Deflection of the beam at point of consideration with

respect to time. ( )nT t Function of time. ( )n x Eigen

functions of the beam (without M) with the same boundary conditions. n= Number of modes of vibration.

Theoretical Study: Considering constant moment of inertia and neglecting the

damping effect, the principal equation of motion of the beam

with moving mass using Euler Bernoullis is given by

(1)

Where = Dirac delta function

Considering the solution of eqn in series form and by variable

separation method, the eqn (1) can be rewritten as

(2)

( )nT t Function of of time or amplitude function to be calculated.

They may be again written as in the form of

4( ) ( ) 0iv x x (3)

Where

The R.H.S of equation (1) can be written in a series form :

(4)

Putting the values of eqn (2) in eqn (4), arranging the

eqn (4) we can write:

(5)

Multiplying both sides of eqn. (5) p(x) and integrating over

Entire beam length

(6)

By using the properties of Dirac Delta function and principle

of orthogonality of p(x)

(7)

(8)

Substituting the values of eqn. (7) and (5) in eq

n. (1),To get the

values of ( )nT t

(9)

Now applying equation (3) in equation (9);

(10)

Now the above equation may be reduced to :

(11)

The equation (11) must satisfy for every values of x and it is

solved by 4th

order Runge-Kutta Method.

Numerical Study: For the numerical analysis the dynamic analysis of the simply supported beam has been carried out

and the CIS value of the corresponding beam with moving

mass has been found out.

Beam Type- Mild Steel, Beam size = (80.350.02) m, n=3

Moving mass= 480 kg.

4 2 2 2 22

4 2 2 2[ 2 ] ( )

y y y y yEI m M g V V x

t xx t x t

0

( ) ( ) ( ) , 0

L

f x x dx f L

0

( ) ( ) 0 , 0

L

f x x dx

1

( , ) ( ) ( )n nn

y x t x T t

42 n

n

EI

m

2 2 22

2 21

[ 2 ] ( ) ( )n nn

y y yM g V V x x S

t xx t

2 // /,

1 1

,

1

1 0

[ ( ) ( ) 2 ( ) ( )

( ) ( )] ( ) ( )

( ) ( )

n n n n tL

n n

o

n n tt p

n

L

n n p

n

M g V x T t V x T t

dx

x T t x x

S t x

2 // /p ,

1 1

,

1

S ( ) [ ( ) ( ) 2 ( ) ( )

( ) ( )] ( )

n n n n tp n n

n n tt p

n

Mt g V T t V T t

V

T t

0

( ) ( )

0,

Lp

n p

V n px x dx

n p

, ,

1 1

( ) ( )

1

( ) ( ) ( ) ( ),1 1

2( ) [

12 ( ) ( ) ( ) ( )] ( )q q t q q tt n

q q

q T tqq

ivEI x T t m x T tn n n n tt

n n

Mx g Vn

Vn

nV T t T t

4 2 //,

1

/, ,

1 1

( ) ( ) [ ( ) ( )

2 ( ) ( ) ( ) ( )] ( ) 0

n n n tt q qn q

q q t q q tt n

q q

MEI T t mT t g V T t

V

V T t T t

4( ) ( ),

2 // /( ) [ ( ) ( ) 2 ( ) ( ) 0,

1 1 1

( ) ( )] ( ),1

MEI T t mT tn n n tt

Vn

x g V T t V T tn q q q q tn q q

T tq q tt nq

2 // /, ,

1 1 1

1

[ ( ) ( ) 2 ( ) ( ) ( )] ( )

( ) ( )

n n n n t n n tt

n n n

n n

n

M g V x T t V x T x T t x

x S t

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Results and Discussion: The CIS may be analysed as the velocity of the traversing mass at which the structure produces

maximum deflection with time variation. The dynamic

response of the simply supported structures of hinged-hinged

& fixed-fixed type with various velocities have been examined

and the deflection at the mid span of the beam also calculated.

It has been observed that for the hinged-hinged type beam

115m/s and for the fixed-fixed type simply supported beam

215m/s is the critical influential speed of the beam. It has been

observed that if we are increasing the speed of the moving

mass beyond the CIS value, the beam deflection will start

decreasing. The consequence of critical speed and the inertia

pertaining to the traversing mass has also been studied. The

displacement of the traversing mass and the structure under

the location of the moving object are calculated in terms of

timet.

Conclusion: The dynamic response of simply supported beam with various velocities at different boundary conditions

has been investigated theoretically and computationally. The

CIS value of the corresponding beam also calculated from the

beam deflection. The deflection of the beam mainly depends

upon the velocities of the moving object.

References:

1. Jeffcott.H ; On the vibration of beams under the action of moving loads,

Philosophical magazine series7. 1929, Vol.8 (48),pp.66-97.

2. M. M. Stanisic, Baltimore, Md., J. C. Itardin and Y. C. Lou, Lafayette, Ind. On the response of the plate to a multi-masses moving system, Acta

Mechanica, 1968,vol.5,pp.37-53.

3.M.M Stanisic and Hardin.J.C,On the response of beams to an atbitry number of moving masses, J.Franklin Institute,1969,Vol.187(2),pp.115-123.

4. M.M.Stanisic,On the dynamic behavior of the structures carrying moving

masses,Ingenieur-Archiv of Applied Mechanics,1985,Vol.55(3),pp.176-185. 5. John E.Akin and Massood Mofid,Numerical solution for the response of

beams with moving mass, Journal of Structural Engg,1889,

Vol.115(1),pp.120-132.

6. M.Olsson, On The Fundamental Moving Load Problem, Journal of sound and vibration, 1991, Vol. 145(2), pp.299-307

7. D.R Parhi and A.K.Behera , Dynamic deflection of a cracked beam with

moving masses, Journal of mechanical engineering science,1997,Vol.211,pp.77-87.

8. M. Ichikawa, Y.Miyakawa and A. Mastuda , Vibration analysis of a

continuous beam subjected to a moving mass, Journal of Sound and Vibration, 2000,Vol.203(4),pp..493-506.

9. S.A.Q. Siddiqui, M. F . Golnaraghi and G.R.Heppler , Dynamics of a

flexible beam carrying a moving mass using perturbation ,numerical ,time-frequency analysis technique, Journal of Sound and Vibration,

2000,Vol.229(5),pp.1023-1055.

10. S. Eftekhar Azam , M. Mofid and R. Afghani Khoraskani , Dynamic

response of Timoshenko beam under moving mass , Scientia Iranica , 2013,

Vol. 20 (1), pp.5056.

11.Cristiano Bilello, Lawrence A. Bergman and Daniel Kuchma ,

Experimental Investigation of a Small-Scale BridgeModel under a Moving

Mass , Journal of Structural Engineering ,2004, Vol.130(5),pp.799-804.

12. K.Y. Lee and A. A. Renshaw , Solution of the Moving Mass Problem

Using Complex Eigen function Expansions ,Journal of Applied

Mechanics,2000,Vol.67, pp.823-827.

13. Pawel Sniady , Dynamic Response of a Timoshenko Beam to a Moving Force , Journal of Applied Mechanics, 2008, Vol.75, pp.24503(1-4).

14. E. Esmailzadeh and M . Ghorashi , Vibration analysis of a Timoshenko

beam subjected to a Travelling mass, Journal of Sound and Vibration,1996, Vol.199(4),pp. 615-628.

15. M. Dehestani, M. Mod and A. Vafai , Investigation of critical inuential speed for moving mass problems On beams, Journal of Applied Mathematical Modelling,2009, Vol.33,pp-

3885-3895.

16. L.Fryba, Vibration of solids and structures (1999), Third Edition, Thomas Telford Ltd , Prague

17. W.T.Thomson, Theory of Vibration with Application (2002) Third

Edition.CBS Publishers , New Delhi.

Timet in Sec(Fixed-Fixed Beam)

Defl

ecti

on

at

mid

spa

n (

mm

)

Def

lect

ion

at

mid

spa

n (

mm

)

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[1]

Mathematical modelling of steady state temperature distribution due to heat loss

from weld bead of a butt joint Jaideep Dutta, Postgraduate Research Scholar

Department of Mechanical Engineering National Institute of Technology Karnataka,

Surathkal Mangalore, India

Email-id: [email protected]

AbstractIn this paper a steady state non-linear analytical thermal analysis has been carried out to determine the temperature distribution due to heat loss from weld bead formed in a butt weld joint. The perturbation method has been incorporated to judge the nature of temperature distribution from the weld pool towards the longitudinal direction of the rectangular plate. The analysis has introduced non-dimensional variable in conjunction with variable thermal conductivity. For two different materials, stainless steel of grade AISI 316 and low carbon steel of grade AISI 1005, the temperature distribution phenomena has been observed. The analysis reveals that the non-dimensional parameter of temperature derived from analytical model has shown good agreement with the established model of moving point heat source and this entails that temperature cycle of weld bead due to heat loss by conduction, is in decreasing mode towards the longitudinal direction of the welded plate.

Keywords Perturbation technique, temperature cycle, moving point heat source, non-dimensional parameter.

Nomenclature

Asymptotic function

K Thermal conductivity

T Temperature

Perturbation parameter

x Space variable

Thermal expansion coefficient (/C)

Non-dimensional temperature distribution term

X Non-dimensional length

L Length of the weld plate

Specific Heat (J/kg-K)

Net heat input (J/m)

Narendranath S., Professor

Department of Mechanical Engineering National Institute of Technology Karnataka,

Surathkal Mangalore, India

Email-id: [email protected]

Density (kg/m3)

Welding velocity (m/s)

y Particular location from fusion boundary (m)

t Thickness of base metal (m)

Subscripts

0, 1, 2. 0th order, 1st order, 2nd order respectively.

i initial

o Final

p Peak temperature

m Melting point

I. INTRODUCTION

Fusion welding of metals is a process which comprises heating and cooling cycle. The thermal analysis of fusion welding covers numerous aspects such as the physics and behavior of arc with the formation of weld pool, development of residual stress and distortion in weldments, characterization of heat affected zone (HAZ), metallurgical properties of joints. All these properties are integrated to improve the weld structures in structural applications. Quantification thermal aspect shows immense importance as the source of all the changes such as chemicalmetallurgical phenomena in liquid solid interface of weld pool, liquid-solid phase transformation and variation of relevant mechanical properties, are solely dependent of temperature gradients induced due to appearance of successive thermal cycles [1, 2]. Most prominent analytical model to predict the heat transfer characteristics is Rosenthals model [3] and still it is one of the basic simple model for

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[2]

illustration. Pavelic et al. [4] first proposed the heat source as Gaussian distribution on the surface of the welded plate. Dimensional parameters were first introduced by Tsai N. S. and Eagar [5]. After this appreciable work has been done by Goldak et al. [6] assuming pseudo-Gaussian heat source. Some recent notable works done by researchers such as Hou Z. B. and Komanduri R. [7] analytically derived general solutions of moving point heat source in both transient and stationary state; Araya Guillermo and Gutierrez Gustavo [8] obtained analytical solution for a transient, three dimensional temperature distribution due to a moving laser beam; Elsen Van M. et al. [9] presented the analytical and numerical solution for modelling of localized moving heat sources in a semi-infinite medium and illustrated its application to laser processing materials; Kukla-kidawa J. [10] explored the exact solution of temperature distribution in a rectangular plated heated by a moving heat source obtained by Greens function method; Levin Pavel [11] developed a general solution of three dimensional quasi-steady state problem of moving point heat source on a semi-infinite solid; Osman Talaat and Boucheffa Abderrahmane [12] proposed an analytical model based on integral transform and finite cosine Fourier integral transform to compute 3-D temperature distribution in a solid by moving rectangular with surface cooling; Winczek Jerzy [13] suggested an analytical model of computation of transient temperature field in a half infinite body caused by moving volumetric heat source with changeable direction of motion; Parkitny Ryszard and Winczek Jerzy [14] described analytically the solution of temporary temperature field in half infinite body caused by assuming moving tilted volumetric heat source with Gaussian power density distribution with respect to depth. Motivated by these facts the present analysis has been carried out in order to estimate the non-dimensional temperature distribution from weld bead of square butt joint to the longitudinal direction by non-linear steady state analysis. By assuming variable thermal conductivity, the perturbation technique (asymptotic) has been adopted as it allows to evaluate the approximate solutions which cant be determined by traditional analytical method. For different temperatures starting from melting point of the metal i.e. liquid weld pool, the analysis has been illustrated the temperature cycle as the distance increasing towards the rear end of the weld plate.

II. BASIC IDEA OF PERTURBATION

METHOD

Perturbation method, also known as asymptotic method is a simplification of complex mathematical problems [15]. It is very difficult to solve higher order non-linear differential equations analytically

whereas this method is acceptable in this situations. Though numerical methods are very popular for solving mathematical problems but they consist of error. Perturbation method can be an alternative approach for solving equations with comparatively higher accuracy. The first step of implementation of this theory is to nondimensionalize the governing equation. Then it requires a small parameter with very small magnitude which appears as dimensionless form in the equation. This parameter usually denoted as , is in the range of0 1. Then the non-dimensionalized equation needs to be expanded in an asymptotic nature with the form [16]:

= + + + (1)

Then the assumed equation is substituted into governing equations and equating the terms of identical powers of , gives the formulation of nth order formulation.

III. MATHEMATICAL FORMULATION

In order to implement the perturbation method the basic assumptions of this modelling are: (a) 1-D heat conduction, (b) Temperature dependent thermal conductivity, (c) Steady state heat transfer, (d) No energy generation, (e) Other modes of heat transfer (convection and radiation) have been neglected and (f) No phase change. Based on above assumptions, the heat transfer equation on the domain (Fig. 1) is [17]:

K(T)

= 0 (2)

K(T) = K[1 + (T T)] (3)

The boundary conditions are:

T(x = 0) = T (4) T(x = L) = T (5) To identify an appropriate perturbation parameter the following nondimensional variables are used:

=T TT T

, X =x

L

Substituting the nondimensional variables in (2)

{1 + (T T)}

= 0 (6)

Thus the perturbation parameter can be expressed as:

= (T T) (7) Thus (6) becomes,

(1 + )

= 0 (8)

Now as (8) is nonlinear, the boundary equations (4) and (5) becomes:

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[3]

Fig. 1. Schematic diagram of square butt joint of two rectangular plates indicating the domain of heat conduction

(x = 0) = 1 (9) (x = L) = 0 (10)

Now using (1) asymptotic equation as mentioned, (8) results:

{1 + ( + +

+ )}

( + +

+ ) = 0 (11)

After differentiating and expanding, (11) becomes:

+

+

+

+ [

] +

+ 2

+

= 0 (12)

Now equating the identical powers of , yields:

:

= 0 (12.a)

:

+

+ [

] = 0 (12.b)

:

+

+ 2

+

= 0

(12.c) Applying boundary condition (9) into (1), (0) + (0) +

(0) + = 1 (13) Equating identical powers of : (0) = 1, (0) = 0and(0) = 0 Proceeding same way, applying boundary condition (10), into (1): (1) + (1) +

(1) + = 0 (14) Again equating identical powers of :

(1) = 0, (1) = 0and(1) = 0 The solution has been derived as follows: (a) 0th order solution: Applying boundary conditions (0) = 1 and (1) = 0 in 12.a,

ddX

= 0

Thus, = 1 X (15) (b) 1st order solution: Substituting as expressed in (15), into 12.b and applying boundary conditions (0) = 0and (1) = 0 gives,

ddX

= 1

Thus, =

(1 X) (16)

(c) 2nd order solution: Substituting and from (15) and (16) into 12.c and applying boundary conditions (0) = 0 and (1) = 0 yields:

ddX

= 2 3X

Thus, =

(2X X 1) (17)

Finally substituting, and into (1) results:

= (1 X) +

(1 X) +

(2X X 1) +

(18)

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[4]

IV. EMPIRICAL MODEL FOR PEAK TEMPERATURE DISTRIBUTION

According to Masubuchi [20], the temperature distribution peak in welded zones, specifically in the Heat Affected Zone (HAZ) and its vicinity can be determined by using (19)

()=

.

+

() (19)

IV. RESULTS AND DISCUSSION

TABLE I. THERMOPHYSICAL PROPERTIES OF MATERIALS [18, 20]

Material Stainless steel (AISI 316)

Low carbon steel (AISI 1050 )

Melting point (C)

1510 1425

Thermal expansion coefficient (/C)

15.9 10 11.7 10

Density (kg/m3) 8000 7872

Specific heat (J/kg-K)

500 481

0.00 0.15 0.30 0.45 0.60 0.750.00

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

No

n-d

ime

ns

ion

al

tem

pe

ratu

re d

istr

i

conference proceeding of iesa-2014.pdf

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